{"id":310,"date":"2023-06-05T15:30:42","date_gmt":"2023-06-05T15:30:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/annuities\/"},"modified":"2026-02-24T23:01:37","modified_gmt":"2026-02-24T23:01:37","slug":"annuities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/annuities\/","title":{"raw":"Savings Annuities","rendered":"Savings Annuities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate the balance on an annuity after a specific amount of time<\/li>\r\n \t<li>Calculate interest earned and amount deposited in an annuity problem<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Savings Annuity<\/h2>\r\nFor most of us, we aren\u2019t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a <strong>savings annuity<\/strong>. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\"><img class=\"aligncenter size-full wp-image-737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\" alt=\"Glass jar labeled &quot;Retirement.&quot; Inside are crumpled $100 bills\" width=\"640\" height=\"560\" \/><\/a>\r\n\r\nListed below is the savings annuity formula.\r\n<div class=\"textbox\">\r\n<h3>Annuity Formula<\/h3>\r\n[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]\r\n<ul>\r\n \t<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\r\n \t<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\r\n \t<li><em>r <\/em> is the annual interest rate in decimal form.<\/li>\r\n \t<li><em>N<\/em> is the number of years<\/li>\r\n \t<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\r\n<\/ul>\r\nIf the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.\r\n\r\n<\/div>\r\nFor example, if the compounding frequency isn\u2019t stated:\r\n<ul>\r\n \t<li>If you make your deposits every month, use monthly compounding, <em>k<\/em> = 12.<\/li>\r\n \t<li>If you make your deposits every year, use yearly compounding, <em>k<\/em> = 1.<\/li>\r\n \t<li>If you make your deposits every quarter, use quarterly compounding, <em>k<\/em> = 4.<\/li>\r\n \t<li>Etc.<\/li>\r\n<\/ul>\r\n<div class=\"textbox\">\r\n<h3>When do you use this?<\/h3>\r\nAnnuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.\r\n\r\nCompound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.\r\n<ul>\r\n \t<li>Compound interest: One deposit<\/li>\r\n \t<li>Annuity: Many deposits.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall order of operations<\/h3>\r\nUsing the order of operations correctly is essential when using complicated formulas like the annuity formula.\r\n\r\nPEMDAS: First simplify like terms inside parentheses then handle exponents before multiplying or dividing. Do addition and subtraction outside of parentheses last.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nA traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?\r\n[reveal-answer q=\"261481\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261481\"]\r\n\r\nIn this example,\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>d<\/em> = $100<\/td>\r\n<td>the monthly deposit<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>r<\/em> = 0.06<\/td>\r\n<td>6% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>N<\/em> = 20<\/td>\r\n<td>\u00a0we want the amount after 20 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPutting this into the equation:\r\n\r\n[latex]\\begin{align}&amp;{{P}_{20}}=\\frac{100\\left({{\\left(1+\\frac{0.06}{12}\\right)}^{20(12)}}-1\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left({{\\left(1.005\\right)}^{240}}-1\\right)}{\\left(0.005\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left(3.310-1\\right)}{\\left(0.005\\right)}\\\\&amp;{{P}_{20}}=\\frac{100\\left(2.310\\right)}{\\left(0.005\\right)}=\\$46200 \\\\\\end{align}[\/latex]\r\n\r\nThe account will grow to $46,200 after 20 years.\r\n\r\nNotice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is $46,200 - $24,000 = $22,200.\r\n\r\n[\/hidden-answer]\r\n\r\nThis example is explained in detail here.\r\n\r\nhttps:\/\/youtu.be\/quLg4bRpxPA\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?\r\n[reveal-answer q=\"160692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"160692\"]\r\n<div>\r\n\r\n<em>d<\/em> = $5\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 the daily deposit\r\n\r\n<em>r<\/em> = 0.03 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3% annual rate\r\n\r\n<em>k<\/em> = 365 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since we\u2019re doing daily deposits, we\u2019ll compound daily\r\n\r\n<em>N<\/em> = 10 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we want the amount after 10 years\r\n\r\n[latex]P_{10}=\\frac{5\\left(\\left(1+\\frac{0.03}{365}\\right)^{365*10}-1\\right)}{\\frac{0.03}{365}}=21,282.07[\/latex]\r\n\r\nThe account will be worth $21,282.07 after 10 years. How much of that is from interest earned?\r\n\r\nYou deposited $5 per day for 10 years. That's [latex]5\\text{ dollars }\\ast 365\\text{ days } \\ast 10\\text{ years }=18250\\text{ dollars}[\/latex].\r\n\r\nSubtract the amount you deposited, $18,250, from the account balance, $21,282.07. You earned $3,032.07 from interest.\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6691[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div>\r\n<h2>Solving For The Deposit Amount<\/h2>\r\nFinancial planners typically recommend that you have a certain amount of savings upon retirement. \u00a0If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nYou want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?\r\n[reveal-answer q=\"897790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"897790\"]\r\n\r\nIn this example, we\u2019re looking for <em>d<\/em>.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>r<\/em> = 0.08<\/td>\r\n<td>8% annual rate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>k<\/em> = 12<\/td>\r\n<td>since we\u2019re depositing monthly<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>N<\/em> = 30<\/td>\r\n<td>30 years<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>P30<\/em> = $200,000<\/td>\r\n<td>The amount we want to have in 30 years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn this case, we\u2019re going to have to set up the equation, and solve for <em>d<\/em>.\r\n\r\n[latex]\\begin{align}&amp;200,000=\\frac{d\\left({{\\left(1+\\frac{0.08}{12}\\right)}^{30(12)}}-1\\right)}{\\left(\\frac{0.08}{12}\\right)}\\\\&amp;200,000=\\frac{d\\left({{\\left(1.00667\\right)}^{360}}-1\\right)}{\\left(0.00667\\right)}\\\\&amp;200,000=d(1491.57)\\\\&amp;d=\\frac{200,000}{1491.57}=\\$134.09 \\\\\\end{align}[\/latex]\r\n\r\nSo you would need to deposit $134.09 each month to have $200,000 in 30 years if your account earns 8% interest.\r\n\r\n[\/hidden-answer]\r\n\r\nView the solving of this problem\u00a0in the following video.\r\n\r\nhttps:\/\/youtu.be\/LB6pl7o0REc\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]6688[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate the balance on an annuity after a specific amount of time<\/li>\n<li>Calculate interest earned and amount deposited in an annuity problem<\/li>\n<\/ul>\n<\/div>\n<h2>Savings Annuity<\/h2>\n<p>For most of us, we aren\u2019t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a <strong>savings annuity<\/strong>. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-737\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/12\/02171735\/7027606047_cac49c3b79_z.jpg\" alt=\"Glass jar labeled &quot;Retirement.&quot; Inside are crumpled $100 bills\" width=\"640\" height=\"560\" \/><\/a><\/p>\n<p>Listed below is the savings annuity formula.<\/p>\n<div class=\"textbox\">\n<h3>Annuity Formula<\/h3>\n<p>[latex]P_{N}=\\frac{d\\left(\\left(1+\\frac{r}{k}\\right)^{Nk}-1\\right)}{\\left(\\frac{r}{k}\\right)}[\/latex]<\/p>\n<ul>\n<li><em>P<sub>N<\/sub><\/em> is the balance in the account after <em>N<\/em> years.<\/li>\n<li><em>d<\/em> is the regular deposit (the amount you deposit each year, each month, etc.)<\/li>\n<li><em>r <\/em> is the annual interest rate in decimal form.<\/li>\n<li><em>N<\/em> is the number of years<\/li>\n<li><em>k <\/em>is the number of compounding periods in one year.<\/li>\n<\/ul>\n<p>If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.<\/p>\n<\/div>\n<p>For example, if the compounding frequency isn\u2019t stated:<\/p>\n<ul>\n<li>If you make your deposits every month, use monthly compounding, <em>k<\/em> = 12.<\/li>\n<li>If you make your deposits every year, use yearly compounding, <em>k<\/em> = 1.<\/li>\n<li>If you make your deposits every quarter, use quarterly compounding, <em>k<\/em> = 4.<\/li>\n<li>Etc.<\/li>\n<\/ul>\n<div class=\"textbox\">\n<h3>When do you use this?<\/h3>\n<p>Annuities assume that you put money in the account <strong>on a regular schedule<\/strong> (every month, year, quarter, etc.) and let it sit there earning interest.<\/p>\n<p>Compound interest assumes that you put money in the account <strong>once<\/strong> and let it sit there earning interest.<\/p>\n<ul>\n<li>Compound interest: One deposit<\/li>\n<li>Annuity: Many deposits.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall order of operations<\/h3>\n<p>Using the order of operations correctly is essential when using complicated formulas like the annuity formula.<\/p>\n<p>PEMDAS: First simplify like terms inside parentheses then handle exponents before multiplying or dividing. Do addition and subtraction outside of parentheses last.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261481\">Show Solution<\/span><\/p>\n<div id=\"q261481\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example,<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>d<\/em> = $100<\/td>\n<td>the monthly deposit<\/td>\n<\/tr>\n<tr>\n<td><em>r<\/em> = 0.06<\/td>\n<td>6% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>since we\u2019re doing monthly deposits, we\u2019ll compound monthly<\/td>\n<\/tr>\n<tr>\n<td><em>N<\/em> = 20<\/td>\n<td>\u00a0we want the amount after 20 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Putting this into the equation:<\/p>\n<p>[latex]\\begin{align}&{{P}_{20}}=\\frac{100\\left({{\\left(1+\\frac{0.06}{12}\\right)}^{20(12)}}-1\\right)}{\\left(\\frac{0.06}{12}\\right)}\\\\&{{P}_{20}}=\\frac{100\\left({{\\left(1.005\\right)}^{240}}-1\\right)}{\\left(0.005\\right)}\\\\&{{P}_{20}}=\\frac{100\\left(3.310-1\\right)}{\\left(0.005\\right)}\\\\&{{P}_{20}}=\\frac{100\\left(2.310\\right)}{\\left(0.005\\right)}=\\$46200 \\\\\\end{align}[\/latex]<\/p>\n<p>The account will grow to $46,200 after 20 years.<\/p>\n<p>Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The difference between what you end up with and how much you put in is the interest earned. In this case it is $46,200 &#8211; $24,000 = $22,200.<\/p>\n<\/div>\n<\/div>\n<p>This example is explained in detail here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Saving Annuities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/quLg4bRpxPA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q160692\">Show Solution<\/span><\/p>\n<div id=\"q160692\" class=\"hidden-answer\" style=\"display: none\">\n<div>\n<p><em>d<\/em> = $5\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 the daily deposit<\/p>\n<p><em>r<\/em> = 0.03 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3% annual rate<\/p>\n<p><em>k<\/em> = 365 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since we\u2019re doing daily deposits, we\u2019ll compound daily<\/p>\n<p><em>N<\/em> = 10 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 we want the amount after 10 years<\/p>\n<p>[latex]P_{10}=\\frac{5\\left(\\left(1+\\frac{0.03}{365}\\right)^{365*10}-1\\right)}{\\frac{0.03}{365}}=21,282.07[\/latex]<\/p>\n<p>The account will be worth $21,282.07 after 10 years. How much of that is from interest earned?<\/p>\n<p>You deposited $5 per day for 10 years. That&#8217;s [latex]5\\text{ dollars }\\ast 365\\text{ days } \\ast 10\\text{ years }=18250\\text{ dollars}[\/latex].<\/p>\n<p>Subtract the amount you deposited, $18,250, from the account balance, $21,282.07. You earned $3,032.07 from interest.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6691\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6691&theme=oea&iframe_resize_id=ohm6691&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div>\n<h2>Solving For The Deposit Amount<\/h2>\n<p>Financial planners typically recommend that you have a certain amount of savings upon retirement. \u00a0If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q897790\">Show Solution<\/span><\/p>\n<div id=\"q897790\" class=\"hidden-answer\" style=\"display: none\">\n<p>In this example, we\u2019re looking for <em>d<\/em>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>r<\/em> = 0.08<\/td>\n<td>8% annual rate<\/td>\n<\/tr>\n<tr>\n<td><em>k<\/em> = 12<\/td>\n<td>since we\u2019re depositing monthly<\/td>\n<\/tr>\n<tr>\n<td><em>N<\/em> = 30<\/td>\n<td>30 years<\/td>\n<\/tr>\n<tr>\n<td><em>P30<\/em> = $200,000<\/td>\n<td>The amount we want to have in 30 years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In this case, we\u2019re going to have to set up the equation, and solve for <em>d<\/em>.<\/p>\n<p>[latex]\\begin{align}&200,000=\\frac{d\\left({{\\left(1+\\frac{0.08}{12}\\right)}^{30(12)}}-1\\right)}{\\left(\\frac{0.08}{12}\\right)}\\\\&200,000=\\frac{d\\left({{\\left(1.00667\\right)}^{360}}-1\\right)}{\\left(0.00667\\right)}\\\\&200,000=d(1491.57)\\\\&d=\\frac{200,000}{1491.57}=\\$134.09 \\\\\\end{align}[\/latex]<\/p>\n<p>So you would need to deposit $134.09 each month to have $200,000 in 30 years if your account earns 8% interest.<\/p>\n<\/div>\n<\/div>\n<p>View the solving of this problem\u00a0in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Savings annuities - solving for the deposit\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LB6pl7o0REc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6688\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6688&theme=oea&iframe_resize_id=ohm6688&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-310\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Annuities. <strong>Authored by<\/strong>: David Lippman. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>Project<\/strong>: Math in Society. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Retirement. <strong>Authored by<\/strong>: Tax Credits. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/flic.kr\/p\/bH1jrv\">https:\/\/flic.kr\/p\/bH1jrv<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Savings Annuities. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/quLg4bRpxPA\">https:\/\/youtu.be\/quLg4bRpxPA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Savings annuities - solving for the deposit. <strong>Authored by<\/strong>: OCLPhase2&#039;s channel. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LB6pl7o0REc\">https:\/\/youtu.be\/LB6pl7o0REc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 6691, 6688. <strong>Authored by<\/strong>: Lippman,David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li> Determining The Value of an Annuity. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/DWFezRwYp0I\">https:\/\/youtu.be\/DWFezRwYp0I<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Determining The Value of an Annuity on the TI84. <strong>Authored by<\/strong>: Sousa, James (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OKCZRZ-hWH8\">https:\/\/youtu.be\/OKCZRZ-hWH8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Annuities\",\"author\":\"David Lippman\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"Math in Society\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Retirement\",\"author\":\"Tax Credits\",\"organization\":\"\",\"url\":\"https:\/\/flic.kr\/p\/bH1jrv\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Savings Annuities\",\"author\":\"OCLPhase2\\'s channel\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/quLg4bRpxPA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Savings annuities - 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